Physica B 169 (1991) North-Holland
Fluctuation, ‘melting’, depinning, creep, and diffusion of the flux-line lattice in high- T, superconductors E.H. Brandt Max-Planck-lnstitut The invited
talk was given
1, D-7000 Stuttgart
by the author.
The transition in crystalline high-T, superconductors from irreversible to reversible magnetic behavior above a field-dependent temperature, first ascribed to ‘glassiness’ then to ‘flux melting’, is caused by thermally activated depinning of vortices. Due to the easiness of vortex cutting and to the presence of many pins per vortex, the nature of the three-dimensional strongly fluctuating ‘vortex liquid’ is not yet known. Thermal depinning is a complicated problem, too, since plasticity of the vortex lattice (dislocations, cutting and reconnection, vortex kinks in the CuO-layers) is crucial for collective flux creep. Very likely, due to vortex crossing and dislocation loops the resistivity is ohmic for current density J-to, i.e., the activation energy is finite and thermally assisted flux flow (TAFF) and flux diffusion occur rather than a vortex glass state.
Glass-like behaviour. Soon after the discovery of high-T, superconductors (HTSC) Miiller et al. [l] observed a change from irreversible to reversible magnetic behavior in these oxides at a fielddependent temperature T,,,(B). The complete disappearance of the hysteresis in the magnetization curve above T,,,(B) is closely related to the observed broadening of the resistive transition near T, in a magnetic field B and to the logarithmic time dependence of the zero-field cooled and remanent magnetizations. These phenomena were first interpreted as ‘glass-like behavior’ in analogy to spin glasses, which exhibit a similar logarithmic time law and a similar reversibility line T, - T,,,(B) m B2’3 called de AlmeidaThouless line. Glassiness initially was suggested by the granular structure of these ceramic superconductors, which was described as a random or regular network of grains connected by Josephson junctions [2-31 or as an effective medium . The length scale for this ‘glassiness’, i.e., the size (fi/2eB)1’2 of the Josephson loops required to
explain these experiments, however, was found to be much smaller than the usual grain size of typically 10 km. Later, when single crystals could be produced from HTSC, most of the ‘glassy’ phenomena were also observed in these crystals. In order to save the glass interpretation one had to assume some ‘initial glassiness’ of grains or single crystals. In order to save the glass interpretation one had to assume some ‘internal glassiness’ of grains or single crystals, perhaps twin boundaries (in caused by the or large YBa,Cu,O,_,) or by the CuO-layers lattice cell . Thermal depinning. Eventually it became clear that the reversibility transition in single crystals can be explained just as in classical type-II superconductors by the presence of Abrikosov vortices and their thermal activation. Such vortices (magnetic flux lines) were observed in HTSC  by the Trauble-Essmann  decoration of the flux pattern at the sample surface with ferromagnetic micro-crystallites (‘magnetic smoke’). At zero temperature, vortices are pinned by material inhomogeneities and will not move under the action of a Lorentz force density J X B if the
E. H. Brundi
of the flux-line
current density J I- /qy’V x B is less than a critical value J,(B = flux density). At T > 0, some of the vortices are unpinned by thermal activation , and the small average drift velocity u of the vortex lattice induces an electric field E = B X u. In classical type-11 superconductors this effect is observed very close to T, in form of flux creep  when the superconductor is in a critical state, i.e., when J=J, or IVxB(=lVBl=pJ,. In HTSC, giantflflux creep [lo-111 occurs in a large temperature interval below T,; even at low current densities J < J, thermally assisted ftux flow (TAFF)  occurs with an ohmic (linear) resistivity  pTAFF. In contrast to the normal resistivity p, (at T > T,) and the usual flux flow (observed at J + J,), the resistivity pFF =p”BIB,, small TAFF resistivity is thermally activated, pTAFF CCexp(- Ulk,T), where CTis an activation energy for flux jumps. TAFF explains [lo-161 the reversibility transition, the field-caused broadening of p(T), and the giant flux creep in a consistent way without introducing new concepts. Vortex lattice melting. After the glassiness and giant flux creep concepts, the idea offlux melting emerged as a possible explanation of the reversibility transition [17, 181. It was suggested that in HTSC the vortex lattice (VL, now often called flux lattice) ‘melts’ when the thermal fluctuation of the vortex positions (u’) becomes large, (Us)“‘= ca, where a -(QO/B)“’ is the c =O.l (Lindemann vortex spacing and criterion). This concept has proved useful for two-dimensional (2D) vortex lattices in films, where a Kosterhtz-Thouless transition is observed . The fluctuation of the 3D VL in HTSC, first estimated by Nelson and Seung [ 181, becomes larger when the elastic non-locality and material anisotropy are considered [20-221, see section 3. The nature of a possible melting transition and the properties of a 3D ‘vortex liquid’, in particular the measurable consequences, are not clear at present: an improved theory should account for the correct interaction between curved vortices and allow for easy cutting and reconnection of vortices and for pinning by material inhomogeneities, see figs. l(a)-(d).
lattice in high-T,
Does melting reduce pinning? It has been assumed, explicitly or tacitly, that ‘a rigid VL is pinned by even a few pins, but a melted VL cannot be pinned’ since it flows between widely spaced pins, see fig. l(c). This seemingly obvious conclusion led to the interpretation of experiments with oscillating superconductors as ‘evidence for flux melting’ . However, as shown by Esquinazi , the observations in ref.  (a sharp peak in the damping and the disappearance of the field-caused enhancement of the mechanical resonance frequency at the melting temperature, see fig. l(k)) can be explained quantitatively by thermally activated depinning: the melting line T,(B) coincides with the reversibility line T,,,(B) measured on the same materials by resistivity and magnetization methods. Due to the diffusive nature of flux motion in the TAFF regime, these lines depend on the measuring frequency v and specimen size L. Reversibility sets in when L or v coincide with the skin depth or reciprocal time for flux penetration, which both increase exponentially with T, see section 7. No: There are many pins per flux line. Weakening of pinning by flux melting would occur if one had much less pins than flux lines. In 3D HTSC, however, each flux line is pinned by many small pins, e.g., by oxygen vacancies, see fig. l(d). Therefore, even a ‘gas’ of non-interacting flux lines would be strongly pinned. The basic idea of statistical summation of pinning forces is just that a soft VL is stronger pinned than a rigid VL [24,25]. An ideally rigid VL cannot be pinned at all since the forces of randomly positioned pins average to zero when the vortices cannot adjust to the pin potential. One might thus argue that pinning should increase at the melting temperature rather than vanish. This would be an oversimplification, too, since the effect of VL softening has to compete (a) with thermally activated depinning [8-161 and (b) with thermal smoothing of the random pin potential  which enhances the range of the pins rP = 5 to rP = ([* + ( u~))I’~ and thereby
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E. H. Brandt
of the flux-line
when B is in the a-b-plane since now A, = PA,, and 5, = .&+lp enter with p 9 1 (p = 5 for YBCO), (b) when the VL contains structural defects or (c) is pinned (pinning reduces ch6! w en some of the gaps between the ) and (d) h pins are wider than assumed here. Conclusion: These arguments show that the indirect pinning by neighboring vortices, which could be reduced by VL melting, is much smaller than the observed direct pinning of each vortex by many pins along its length.
lattice in high-T< superconductors
distance and shear strain ) one would obtain a melting temperature T,, = tT, which is implicitly given by [21,22, 301 t/(1 _ [a)“2 = t*(l _ ,)“*/b”,
t* = T* IT,, T* = [email protected]
&,kBA,(0)K. A,(T,) 0~(1 - t4)-“2. With used We B,,( T,) x (1 - t2) this gives for B 4 Bc2,
T,,,(B) = [email protected]
;‘2/~0k,A~,(T,)l-B”2 3. Does a pin-free
vortex lattice melt?
Thermal fluctuation of the vortex positions as obtained from linear elasticity of the VL [7,21,27] is
(4) (u’> = k,T(
Here BZ means the Brillouin zone of the VL of k; = area Tk&, kiZ = [email protected]
“, b = B/B,__(T), kz + kt,, and cd4(k) is the tilt modulus of the VL in a uniaxially anisotropic superconductor with B]]z](c-axis at B>2B,, . The first factor in eq. (5) is Nelson’s  result of local elasticity theory [using cd4(k = O)]. The second and third factors are large corrections due to nonlocal elasticity [using ca4(k) in eq. (4)]  and to material anisotropy . For YBCO one has ~200 and A,.lA,, = K = A,,/&, r=5, and bK2/(l - b) = 2nh2BlQ0(1 - b) = 3BIB,,(l - BIB,,) 9 1. From Lindemann’s criterion (u’) ‘I2 = ca and from other melting criteria (fluctuating vortex
which means B(T,) m (T, - T,)’ near T, and B( T,) CCli Ti far from T,. The thus defined melting temperature T,(B) is rather low; one has T” = 40 K for YBCO and c = 0.1. Softening of the VL by structural defects or nonlinear elasticity decreases T, even more. It is not clear at present whether there is an abrupt melting transition in a 3D VL. The always present pinning destroys long-range order in the VL even at T = 0 , and thermal fluctuation might just increase the disorder continuously. But even in an idealized, pin-free VL the nature of the liquid state is not yet known since in ref.  a simplified interaction between nearly parallel vortices of distance r L was used, V(rl) 0~K,(r L/A) (K,, is a modified Bessel func[ion), see fig. l(a). This, too rigid, interaction has a large barrier for vortex cutting or crossing [K,,(x) = -In(x) f or x < 1 yields a repulsive force = l/ rl] and suppresses fluctuations with vortex curvature of wavelengths <27rA = (5Bi B,,)“2a 9 a. Such fluctuations and vortex cutting are facilitated by the correct 3D interaction, even more by the anisotropic interaction (section 4), and still more by the layered structure of HTSC (section 5).
The anisotropic London theory applies to all temperatures 0 < T < T, if the vortex cores do not overlap (i.e., for B <0.25B,,, otherwise the Ginzburg-Landau theory has to be used) and if the coupling between the superconducting layers
E. H. Brandt I Properties of the flux-line lattice in high-T, superconductors
is not too weak (i.e., for layer distance s < .&, else the Lawrence-Doniach model of section 5 should be used). In the presence of vortices the London equation for B(r) reads B+Vx(A.Vxa)=~~~~dr;S(r-r,),
Vortex cutting. Equations (lo)-( 13) apply to arbitrary arrangements of curved or straight vortices, even to loops and crossed vortices. From eq. (13) the interaction between two straight vortices tilted against each other by an angle CY and with separation d[fig. l(b)] becomes  Ui”,(d, (y> = <@i cot cw/2m,A) exp(-d/A)
where the line integral is along the ith vortex position ri(z) = [xi(z); y,(z); z] and A is a tensor. For uniaxial superconductors one has AaP = A,6,, + Azc,cp where col are the Cartesian components of the unit vector 2 along the c-axis (a, p = x, y, z) and A, = Ai, and A2 = h: - hih. The general solution of eq. (9) is 
fi 1 + A,k* + A2q2 > 87r3
with q = k x C. The free energy, with density (2~~))~ [B* + (V x B)A(V x B)], may be written as a tensorial interaction between pairs of line elements dri, which may become attractive even when dri]]drj ,
For isotropic superconductors (A, = A*, A2 = 0) eq. (11) becomes f,,(r) = aaP exp(-r/h)/ 47rA’r. This means, the source field is along drj and rotationally symmetric, see fig. l(f)-(g), and the interaction becomes vectorial, F=
2 c $dr, 0 i i
with r = (ri - rj]. The terms i = j in eqs. (12) and (13) give the self-energy of a vortex; they diverge in this London limit unless a finite vortex core is introduced, e.g., by replacing I by (Y* + 4 57’2,
This does not diverge at zero distance d, even when t-, 0. Finite 5  and spontaneous bending of vortices  reduce the barrier for vortex cutting further.
5. Vortices in layered superconductors A powerful phenomenological theory of superconductors is the Lawrencelayered Doniach model . This modified GinzburgLandau (GL) theory follows from a tight-binding approximation. It replaces the continuous complex order parameter +(x, y, t) by functions $,,(r) belonging to layers positioned at z, = ns; n = integer, r = (x, y), and z is along C. The Lawrence-Doniach free energy functional reads, for layer distance s < A,,
(15) where (Y and p are the usual GL coefficients, B = curl A at z = z,, Z,(r) = (2elh) s,:“” A,dz, A, = .iA, and m and M are effective masses related to [,‘, = h2/2ma, 6; = fi212Ma, Atb = mj3/4,uu,e2a, and A: = MP/4poe2a, thus A,l A,, = .&,I.$, = (Mlm)1’2 + 1. Numerous useful results were derived recently from eq. (15), both for vortices perpendicular [36-381 and parallel [39,40] to the CuO-planes. A few points shall be mentioned here. For .& > s the difference in eq. (15) may be replaced by the gradient a a/az - 2ieA,lfi; this yields the anisotropic GL theory. In the opposite limit, &,@ S,
E. H. Brundt
of the flux-line
eq. (15) describes weakly Josephson coupled superconducting layers. If B IIt, the point vortices in each layer interact with vortices in the same and in other layers magnetically and by Josephson coupling of the phases 4,!(r) of the $,[(r). Both interactions follow from the phasedependent part of eq. (15) (I+,,\‘= ai/3 for B < BC)’
x [I- cos(4,+, - 4, -
For 5, > s, eq. (16) yields the anisotropic London theory, eq. (9); the singularities on the RHS of eq. (9) follow from a term (@“/27~)V X (V$) and the 3D vortex phase 4. For 5,. < s the point vortices in adjacent layers [fig. l(h)] interact mainly magnetically at distances up to the where A, = s(M/m)“2, length Josephson F o(ln(r/A, ). At r > A, Josephson coupling m cos(+,, + , - $,1) dominates. As discussed in [37,38], melting and pinning of the VL in the layers is 2D when the inter-layer coupling of vortices is weaker than k,T or than the pinning energy. In the limit of zero coupling (M/m-+ =. A, -+ m), ‘pancake vortices’  result, hJ+m, see fig. l(h): the field lines of a point vortex are screened by the other layers and compressed into a layer of thickness 2h,,. Point vortices have B along 6 and, when stacked, exhibit only a small tilt stiffness of order chh [cf. A, = 00 inserted in c,,(k) in eq. (4)]. A finite Josephson coupling of the layers is required to allow for tilted B and for currents along i; with non-zero coupling, the large tilt modulus ~~~(0) = BH = B*/p,, for uniform tilt obtains from eq. (4) at long wavelengths >27rA<. For B nearly along the layers, vortex kinks [fig. l(i)] [40,42] become important for both the anisotropic magnetization (which depends on the kink number x ( sin 41 and thus non-analytically on small tilt angles 4) and for thermal depinning (which crucially depends on the type of the
in high-T< superconductors
thermally activated, formed by vortex kinks
The theory of thermally activated depinning still suffers from the fact that the physical meaning of the activation energy U, jumping volume V, and jump width 1 is not known for real superconductors. According to Anderson  vortex jumps along and against the Lorentz force JB, with rates CI:exp[-(U -t W)/k,T] where W= JBVI, generate a mean electric field [lo12, 161 E(J)
= 2p,J, exp(-
Here J, = UiBVl (critical current density), p, (resistivity at J = J,), and U(B, T) are heuristic parameters. One has three limiting cases: P = PTAFFtLcxP(p mexp[(J/J,
for J % J,k,T/U
- l)UIk,T] for J = J, (flux creep)
P = PFFz P,B’Bc2(T)
for J > J (flux flow) > c
A smooth transition to the usual flux flow regime, e.g. by smoothly joining the curve in eq. (17) at J c JL with E(J)=(J2 - J~!“‘p,, at JS J:_ (J;a J,)  will always exhlblt a negative curvature of E(J) somewhat above J,, see fig. l(j). The figures in ref.  are thus not necessarily indicative of a transition to a vortex glass state. Vortex glass theories  and theories of collective flux creep  predict jump volumes V(J) and activation energies U(J) x Jmfr(a > 0) as J+ 0 and thus yield which diverge p,.AFfzx exp[ I - (J,/J)“] -+ 0 for J+ 0. These theories, however, consider only elastic deformation of the VL. It is likely that plastic deformation and vortex cutting lead to a finite U for J-0 and thus to a finite pTAFF, which is also confirmed experimentally  and by physical arguments [40,46].
E. H. Brandt
I Properties of the flux-line lattice in high-T< superconductors
7. Flux diffusion In the ohmic TAFF and flux-flow regimes the vortex motion is a linear (and for flux creep a nonlinear) diffusion [12, 14, 161. For a rather general case the equation of motion for B(r, t) is
[1613 B =v X[D(B,T)K2B X[B x[Vx II]]],(18) simple For where D(B, T) = P(B, T)//-$,. geometries and in linearized form (when B = const.) eq. (18) reduces to the linear diffusion equation k = DV2B with diffusion constant D = PIAFFIP,, mexp(-Ulk,T). Due to flux diffusion the dissipative part of the AC susceptibility of a HTSC slab of thickness d in parallel field is maximum when the circular frequency w coincides with the reciprocal penetration time for flux-density changes 7-l = Dn”ld’. At this frequency the skin depth 6 = equals d/fin. The attenuation T(T) (D/~w)“~ of a HTSC performing tilt vibrations has a sharp maximum at a T,,,(B) which depends on specimen size and geometry. More geometries are treated in refs. [12, 161. In general, any change in the applied field or current changes the boundary condition for the VL. This causes the VI, to diffuse freely until a new equilibrium or stationary state is reached. References 111 K.A.
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E. H. Brandt
of the ,fiux-line
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